Consider the following argument:
Using / to separate premises, and // to separate the conclusion from the premises, a translation of the argument would be:
A truth table for the entire argument would be:
Note that there are 2 simple statements involved: H and A. Thus there are 4 possible combination of T's and F's (2 X 2), and thus 4 rows in our table. (If you need to review how such tables are made, click here to return to the relevant section of Class III.) Remember our definition of a valid argument: An argument is valid if it is impossible for all the premises to be true, while the conclusion is false. Is it possible for all the premises to be true and the conclusion false in this argument? If you find such a row, click anywhere on it.
Is Dad reasoning correctly? Dad's argument followed by its truth table ("C" is "Bobby can have the cookies", and "I" is "Bobby can have the ice cream". If you need a refresher from Class III on how to translate "not both", click here and look at (1)):
In what rows are both premises true? (3) Is the conclusion true also in those rows? (yes) Therefore the argument is valid. Notice that we only look at the columns below the major connective, the ones marked with an asterisk. Consider this argument:
There are 3 simple statements in this argument. (L, E, G). Therefore, there are how many rows? Eight: 2 X 2 X 2. (A way of making sure you have all 8 combinations: make the first column 4 T's, 4 F's; the second, 2T's, 2F's, etc., and the last, alternate. If you have 4 simple statements, then you have 16 rows (2 X 2 X 2 X 2) If you have 16 rows, it's 8 T's, 8 F's; then 4 at a time, then 2 at a time, then alternate. If you need to see how this table was constructed click here.)
In what rows are all the premises true? (Note the arrows.) Is the conclusion true also in every case? (Yes) Therefore, the argument is valid. Summary: The steps in using a truth table to determine the validity of an argument: 1. Symbolize the argument. 2. Determine how many simple statements there are, and thus determine the number of rows you'll need. 3. Fill out the table as we did above. 4. Check the truth values under the major connectives, and look for the rows in which all the premises are true. 5. If the conclusion is true in all those rows, the argument is valid. If the conclusion is false in one of those rows, it is invalid.
Interactive Examples
Translate the following argument. Use "&" for , ">" for , "-" for , and "=" for . Put a / between premises, and // between premises and conclusion. Don't use any extra spaces. Type your answer in the white spaces below each argument.
Use F and S as your simple statements:
Check your results by clicking here.
Use R, W, and C as your simple statements:
An argument is invalid if it possible for the premises to be true, while at the same time, the conclusion is false. In other words, if you can find one row of a truth table which has true premises and a false conclusion, then you know the argument is invalid. Look at the following argument:
(E v F) / ( E F) (G H) / H G // G
A complete truth table would take 16 rows. However, we suspect that the argument might be invalid. Let's see if we can make the conclusion false, and the premises true. 1) The first step would be assign to G, our conclusion, the value false, or F. Thus, G will be assigned F wherever it occurs. (Note: I'll use red letters for the assignment to simple statements, green for intermediate steps, and blue for the value of the whole statement.)
(
E
v
F)
/
F
)
G
H
//
2) Next we will try to make our premises true. Since G is false, and we want H G to be true, H would have to be false:
T
Notice that if G and H are both false, G H would be false. Let's put an F under the dot to indicate that this conjunction is false:
3. Since the consequent of the second premise is false, and we want the second premise to be true, the antecedent ( E F) must be false. Notice that there are several ways (3 in fact) to make ( E F) false. Let's pick one such way, letting E be true and F be false. This gives us:
4. Is our first premise true? (if it isn't, let's go back and see if we can make any changes. We did have a choice in the second premise.)
Notice that a F v T statement is true. Thus we have true premises and a false conclusion. We have found one line (out of 16) in which the premises are true and the conclusion false. This happens when E is true, F is false, G is false and H is false. There may be other assignments of truth values in which this happens (there is actually one in this case), but we only need one such assignment to prove that this argument is invalid.
One more example. Look at the "Bad Bart" example at the end of this class by clicking here, and then returning using the "back" button. Let's symbolize Bart's argument:
S L / L // S
Let's make the conclusion, S, false:
S
L
For the premises to be true, L is true (obviously):
Since F T is true, we have true premises and a false conclusion:
Thus Bad Bart's argument is invalid. (Of course, he still has a gun, but that's another problem.)
This method of proving an argument invalid is called the "indirect" method. It works fairly well when there are one or two ways in which the conclusion can be false. When there are several ways for the conclusion to be false (e.g., when the conclusion is a conjunction), the method is unwieldy.
This method can also be used to prove that an argument is valid. If after assigning the conclusion the value false, you are led, by necessity, into having to assign both true and false to the same statement to get the premises true (which is impossible) then you know it is impossible for the premises to be true and the conclusion false. The indirect method is most often used, however, when you suspect that the argument is invalid.
As it happens, a great many of the arguments which people present appear in a small number of forms, or patterns. If these common argument forms are invalid, they are called fallacies. If these invalid argument types are identified in terms of their form , as opposed to their content or the meanings of the words used, they are called formal fallacies. For example, look at the following dialogue, again between Dad and Bobby:
Is Bobby reasoning correctly?
Since in row four there are true premises and a false conclusion, this argument is invalid. Since this is a common way of reasoning, this is a fallacy. It is called the fallacy of denying a conjunct. After all, the reason that Bobby can't have the cookies also, may be because he can't have either one of them--look at this possibility in the table. (He may be upset for awhile, but he'll reason better because of the experience.) Following are some common valid argument patterns:
Note that the first Bobby - Dad discussion is an example of the (valid) conjunctive argument. Following are some invalid patterns, or formal fallacies.
Note that the second Bobby-Dad exchange was an example of the fallacy of denying a conjunct. What fallacy is committed in the following argument? (Click here if you aren't sure.)
Can you find Bad Bart's problem in the table of fallacies?
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